In a series of papers, Jafarov, Nagiyev, and their collaborators studied several variants of harmonic oscillator models in a PDM background [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}. They directly solved the corresponding Schrödinger equations and proved that their solutions go over to the well-known ones of the constant-mass harmonic oscillator in some appropriate parameter limit. In a recent work [10]}, they considered a harmonic oscillator potential well with a PDM that has the property of becoming infinite for some negative value \(-a\)  of the position while going to a constant for a large positive value of the latter. This resulted in a semi-infinite quantum well, whose spectrum was determined by a direct approach.
